A THREE DIMENSIONAL EXAMPLE

Consider a quantum particle in an impenetrable cubical box of side length L. We have standing probability waves in each dimension, so we have three principal quantum numbers, n_x, n_y and n_z, one for every degree of freedom of the system.

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DEGENERACY
A quantum system is said to possess degeneracy if the same energy corresponds to two or more different sets of quantum numbers. In physics, degeneracy generally occurs because the system has a symmetry. In the case of the cubical box, the system is symmetric under re-labeling of the coordinate axes. Therefore the states [1,1,2], [1,2,1] and [2,1,1], for example, have the same energy.

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INTRINSIC SPIN: Experiments revealed that fundamental particles, such as the electron, have an intrinsic magnetic moment, even though they are point particles. This moment is usually described in terms of an intrinsic quantum number called the spin. This quantum number, s, is a characteristic of the type of particle, and never varies. The electron has s = 1/2, and the associated magnetic moment has only two possible directions in space, unlike a classical magnetic moment, which is a vector that can point in any direction. As another example, the photon has a spin of 1, but no associated magnetic moment, since it does not have a charge.

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In 1927, P. A. M. Dirac figured out how to write an equation for the electron that was consistent with both quantum physics and relativity. To his astonishment, it included spin automatically... basically, relativity required spin to exist. But even more astonishingly, there were two solutions. In other words, the equations simultaneously described two different particles, the electron and its anti-particle, soon called the positron. Again, relativity requires the existence of a counterpart to every known particle, and a whole new universe of “antimatter.”

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Spin plays another key role in nature. The spin of a particle determines how it behaves when placed in a confined space. Particles with half-integer spin are called fermions, and obey an Exclusion Principle. No two identical fermions can be placed in the same state in the same region of space. Each identical fermion in a system must have a different set of quantum numbers than any other one in the same system. Also, fermions are conserved in number. Particles with integer spin are called bosons, and such particles do not obey an Exclusion Principle. Any number of identical bosons can be placed in the same state in the same region of space. Also, bosons are not conserved in number. Bosons can appear or disappear at any time, as long as conservation laws are obeyed. And if they appear only for a very short time, even energy conservation does not have to apply, since Δ E Δ t ≃ ℏ. Such very short-lived bosons are called virtual.

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Fermion particles and antiparticles can only be created in pairs. Conservation laws require that if you create a particle, you must also create the corresponding antiparticle in the same process. Thus physicists speak of pair creation, and pair annihilation.

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